A common technique, referred to as the Bayesian approach in the optimal design of experiments [1]} is to quantify performance using functions of \({\mathbf {C}}_{{\mathbf {e}}}({S})\)  [2]}.
In particular, Bayesian determinantal or “D”-optimality entails minimizing \(\log {\det {{\mathbf {C}}_{{\mathbf {e}}}({S})}}\) , which, under Gaussian assumptions, is equivalent to minimizing the conditional entropy [3]}, [4]} or the volumes of confidenece ellipsoids about the maximum a posteriori (MAP) estimate of \({\mathbf {g}}({\mathbf {x}})\)  given \({\mathbf {m}}_{{S}}({\mathbf {x}})\)  [5]}.
This approach is widely used for sensor placement since it readily admits efficient approximations based on convex relaxation [5]} and greedy algorithms [7]}, [8]} with guaranteed performance.
Similar objectives have been used to quantify observability and controllability for sensor and actuator placement in linear dynamical systems [9]}, [10]}.
